Development and performance evaluation of SCS-CN based hybrid model

In urban hydrology, estimation of runoff from a watershed is a major activity and fundamental to a range of hydrological applications (Wang et al. 2012), such as design of hydraulic structures, design and planning of soil and water conservation works, assessment and planning of flooding hazards, assessment of non-point source pollution, water resources management, study of reservoir sedimentation at the downstream, etc. (Singh et al. 2010; Chen et al. 2018; Rabori & Ghazavi 2018). During the transformation of rainfall into runoff, there are several variables such as rainfall amount and its pattern, land moisture status with prior rainfall, soil infiltration capacity, watershed slope, land use, etc. that affect the depth of produced surface runoff. Different models exist in the literature and study the outcome of different variables on runoff generation. These models mimic the rainfall-runoff transformation process as per the design and development of the model. Under the law of parsimony, the researchers preferred a rainfall-runoff model that require least input parameters with better runoff prediction. Between them, Soil Conservation Service-curve number (SCS-CN) method (presently also known as the Natural Resources Conservation Service curve number (NRCS-CN) method) is most commonly and widely used method across the globe and maintained by ample documentation (Chung et al. 2010; Singh et al. 2010). This is an event-based model since it estimates runoff without acknowledging base flow component (Santikari & Murdoch 2018). This model is also useful to analyze the impact of urbanization or land-use change on runoff (Li et al. 2018) and estimation as well as verification of CN values.

Due to its simplicity, universally acceptance, and requirement for fewer parameters, the model has enjoyed a long history of its applications. Though the method is attractive and widely applied, the model can be upgraded by circumventing the abrupt jumps in runoff estimation that are possible due to different antecedent moisture conditions (AMC) criteria. The application of this model on dissimilar watersheds having different physiognomies could be inappropriate due to its empirical basis. Despite its extensive use, the NRCS-CN model has several major limitations, such as strong dependency on a single parameter CN, ambiguous effects of the AMC, constant initial abstraction ratio as 0.2, failure to include spatial effects, watershed size limitations, and its variable accuracy in different watersheds (Ponce & Hawkins 1996; Hawkins et al. 2009). This event-based model is very flexible and easily applicable to ungauged watersheds and requires two parameters only, i.e. initial abstraction coefficient (λ) and the potential maximum retention (S) expressed in terms of curve number (CN) for estimating event runoff. The CN value incorporates different factors on its own that reflect various characteristics of watershed and can be adopted for different climatic conditions (Sahu et al. 2007). The CN is a variable and dimensionless value ranges from 0 to 100, and it is an index of hydrologic soil group, land use, land cover, soil surface condition, and AMC.

The runoff prediction capability of the SCS-CN model was very poor for the CN values obtained from the National Engineering Handbook Section 4 (NEH-4) table while its value obtained from the measured dataset gives a better prediction. The hydrological systems are overdesigned using tabulated CN values. These tabulated values originally developed after long experimental work done on small and medium-size US watersheds only. It has been found that it is superior to use the CN value calculated from storm events rather than to select them from NEH-4 tables if measured data is available. To investigate the integrated impact of the rainfall-runoff process on watershed characteristics, the choice of using long-term daily rainfall-runoff data is better rather selecting few events to estimate the CN value (Mishra et al. 2008). The accuracy of the determination method of CN values is unidentified and based on empirical validation. The calibration of CN value is necessary or it will give inaccurate prediction.

Kim & Lee (2008) and Kim et al. (2010) witnessed that the result of runoff estimation was poorer using a CN value directly from the NRCS table without calibration. When the model is applied to the large watershed (size greater than 250 km² or 25,000 ha), extra care should be taken since the spatial and temporal distribution of rainfall becomes critical in such cases (Ponce & Hawkins 1996; D'Asaro & Grillone 2012). If spatial distribution of rainfall is uniform, in such cases watershed size makes no difference. Hawkins et al. (2009) suggested that only large storms (P ≥ 25.4 mm) should be used to minimize biasedness towards high CN values since the SCS-CN model does not work well for very small rainfall events (Cheng et al. 2014). Lian et al. (2020) modified the CN values to reduce runoff errors for Chinese watershed (China-CN) that were different than tabled value.

If the number of parameter increases in a hydrological model, it will increase difficulty in parameter identification particularly in case of ungauged watersheds (Skaugen et al. 2015). In such a case, it becomes very difficult to find better-calibrated values for unidentifiable parameters. So it is necessary and better to formulate or develop models that are parametrically efficient. However, to check the application and mathematical consistency of a model, hydrological scientists have discussed the physical basis and empirical framework of the model (Michel et al. 2005).

There are several advantages of conceptual and empirical models but during recent times, researchers are more focused on the development of hybrid or coupled models to obtain better results. These outcome-based models integrate different prediction models and improve perfection in the final prediction. This improved correctness is achieved by combining more advantages. The main aim to make a hybrid model is to improve their generalization ability, robustness, and improvement of final results. Upreti & Ojha (2021) integrate antecedent rainfall directly in to the SCS-CN model and developed a hybrid model for which runoff predictive efficiency was found to be significantly better. The modification and improvement of hybrid models depend on optimizing the value of different parameters. Due to the optimized parameters used in the model, it becomes less appealing under different climatic settings. There are many models developed in the past that eliminate some limitations and structural inefficiency of SCS-CN model. They improve model structure adding more variables and try to overcome the shortcoming of models.

Since the estimation of the correct CN and λ value is difficult in SCS-CN model, which leads often to poor estimation of runoff, a new hypothesis is considered in this study. In this research paper, a hybrid approach has been used to increase the predictive efficiency of Ajmal model (Ajmal et al. 2015). The model ignored both the initial abstraction and maximum potential retention value of SCS-CN model. Thus, to give conceptual framework of the SCS-CN model, the event-based Ajmal model was merged with the most commonly used SCS-CN method and a new hybrid model was formulated. All parameters of the newly developed model were calibrated using calibration dataset and form a one-parameter model that is very simple and easy to use to predict surface runoff. The constant value of these parameters was determined after optimizing all parameters of this hybrid model. This newly proposed hybrid model is based on a constant (L_c) and rainfall amount (P) to accentuate the variation from storm to storm. The structure of the hybrid model is totally different than the SCS-CN model. To estimate consistency in runoff prediction, this study first examined the SCS-CN model (M_i) and its amendment proposed by Woodward et al. (2003) (M_ii) which use different λ value. Both models were then compared with the Ajmal model (Ajmal et al. 2015) (M_iii) and newly proposed simplified form of hybrid model (M_iv) and all of them were tested on remaining watersheds. The performance of the proposed and earlier developed models was checked using different statistical critera. This study considers normal AMC, which is generally described by AMC II. Therefore, no antecedent rainfall value is required throughout the study.

Thus, due to the change in λ value, the optimized value of S or watershed curve number will also be changed. The present study calibrates CN value by optimization instead of taking it from traditional CN table.

Based on the preliminary assessment and analyses of the Ajmal model (M_iii), this model needed further modification to increase its performance. For this, a new hybrid model is formulated after merging the L_c-based M_iii model with the original SCS-CN model. The proposed hybrid model (M_iv) takes the conceptual framework of the SCS-CN model with the maximum possible retention value S and considers initial abstraction I_a. To meld both models and combine them in a single equation, it is required to rewrite both equations again in a different manner.

Equation (11) shows the general form of a new conceptual-cum-hybrid model. The maximum potential retention value S of this conceptual-cum-hybrid model is different than the actual SCS-CN model due to consideration of model constant L_c.

In the proposed hybrid model, the most appropriate value for all three variables can be found with optimization by minimizing the sum of the squared difference between calculated and observed runoff and using the least square fitting technique for all 78 calibration watersheds. In this model, the initial estimate of parameter S, L_c, and λ were taken as 250, 0.001, and 0.01 respectively which were assumed to vary in the range of 1–5000, 0–1, 0–1 respectively. The range of parameter values of S, L_c and λ for calibrated watersheds by using Equation (11) are presented in Table 2. Due to the variation in mean and median value for all three variables, the median value was preferred over the mean because it represented well for the maximum number of watersheds. The median value of S, λ, and L_c for the calibration dataset were found as 26.05, 0.023, and 0.024 respectively and for simplification these values considered as 25, 0.02, and 0.025 respectively.

To simplify the proposed hybrid model, out of three, two parameters can be removed and replaced with a constant and most appropriate value. The decision of which parameter is to be removed is based on their sensitivity to runoff. To find out the most sensitive parameter, sensitivity analysis was performed.

The aim of sensitivity analysis is to estimate the rate of change in model output with respect to the change in model inputs. It helps to decide the rank of the parameter according to their importance. A less important parameter can be removed to make a simple and easier model while the most influencing parameter requires more attention and accurate value. In this study we perform local sensitivity analysis, which is also known as ‘one-parameter-at-a-time’ sensitivity analysis. Using numerical approximation, we determine the most sensitive parameter among the three variables (S, L_c, and λ) that affect the runoff most at a single rainfall value. To estimate the impact of the most sensitive parameter, the progressive changes of all variables on runoff were examined. One variable was increased or reduced progressively by 5% in each step from its representative value, while other two variables were considered constant (approximate median value). By using rainfall value as 1 inch (25.4 mm), corresponding runoff values were computed using Equation (11). This procedure was implemented one by one for all three variables. After increasing the value of all variables one by one by 5% in each step, the runoff amount is calculated. The calculated runoff amount in each case and runoff variation between them is shown in Figure 1.

The result of sensitivity analysis clearly shows that the L_c is the most sensitive parameter to runoff followed by S and λ. Since L_c is the most sensitive parameter, it is necessary to find the correct value of L_c for each watershed. This value can be obtained by optimization techniques. The resulted value of L_c for each watershed will be different and based on their recorded P-Q value.

It should be noted that the equation of the suggested model is not in the form of the original SCS-CN equation but it retained the easiness and usefulness of the model and required only L_c as a single optimized parameter and can be applied to the ungauged watersheds for estimation of runoff.

This hybrid model requires the least input parameter and follows the law of parsimony. The Equation (12) is the simple form of a conceptual hybrid one-parameter non-linear (M_iv) model. In this equation, we do not need any value except rainfall and optimized L_c for a particular watershed. It is tested on remaining 36 US watersheds and the results were compared to those obtained from the SCS-CN model (M_i), Woodward et al. (2003) model (M_ii) and Ajmal et al. (2015) model (M_iii).

To check the model performance, data in the present study are taken from the United States Department of Agriculture-Agricultural Research Service (USDA-ARS) water database. This water database is a collection of rainfall and stream flow data from small agricultural watersheds of the USA. A total of 114 watersheds with areas varying from 0.17 ha to 30,351 ha were selected. All these watersheds contain a total of 28,849 rainfall-runoff events.

Due to the variation in number of events in each watershed, all 114 watersheds were arranged according to number of events and then watersheds separated for calibration and validation. Based on orderly events for every ten watersheds, at least seven were selected for calibration and the remaining three for validation. In this way, we get better distribution of watersheds according to rainfall events and out of 114 US watersheds, 78 watersheds (7,817 events) were used for calibration and remaining 36 watersheds (3,967 events) for validation to develop the hybrid model.

In parameter estimation, model parameters are optimized systematically corresponding to a selected objective function. The twofold purpose of parameters estimation is to find a conceptually unique and realistic parameter set and to obtain a parameter set that yields the best possible agreement between model-simulated and observed runoff.

In the optimization procedure, all four models M_i, M_ii, M_iii, and M_iv allowed variation in model parameters except rainfall P. The M_i and M_ii model was allowed in variation of parameter S whereas the M_iii and M_iv model was allowed in the variation of parameter L_c. The initial estimate of parameter S was taken as 125 mm for M_i model and 250 mm for M_ii model which were assumed to vary in between 1 and 5,000. Similarly, initial estimate of L_c was taken as 0.001 and was assumed to vary in range of 0–1 for M_iii and M_iv model. After applying all four models, the range of parameter S (M_i and M_ii model) and L_c (M_iii and M_iv model) for all watersheds are presented in Table 2.

The smaller RMSE value indicates better model runoff prediction while higher value is a case of poor performance. It's zero value indicates a perfect fit. McCuen et al. (2006) found NSE as a good criterion for model comparison. The value of NSE is less than or equal to 1. For perfect agreement, NSE value should be 1. If NSE > 0.5, the model is considered to have satisfactory performance (Moriasi et al. 2007). According to Ritter & Muñoz-Carpena (2013) criteria, a model is said to be unsatisfactory if NSE is less than 0.65. Other ratings were acceptable (NSE = 0.65–0.8), good (NSE = 0.8–0.9) and very good (NSE > 0.9). The PBIAS value shows whether the model is consistently overestimating or underestimating. The PBIAS value as zero shows perfect agreement. Positive bias results indicate model underestimation and vice-versa. According to Archibald et al. (2014), PBIAS ≥ ±25% indicates unsatisfactory fit; ±15% ≤ PBIAS < ±25%, satisfactory; ±10% ≤ PBIAS < ±15%, good; and ±10% indicates a very good fit. The higher n(t) value imitates the model appropriateness for efficient runoff computation. If n(t) < 0.7, the model performance considered unsatisfactory. As per Ritter & Muñoz-Carpena (2013) criteria other rating 0.7 ≤ n(t) < 1.2, 1.2 ≤ n(t) < 2.2, and n(t) > 2.2 are considered successively as satisfactory, good and very good.

The model performance can be evaluated using the relative ranking and grading. The ranks (I-IV) were assigned to the models based on their NSE values (Verma et al. 2017), i.e. maximum NSE gives I rank and minimum NSE value assigns IV rank. On the basis of their rank, grade points 4, 3, 2, and 1 are given to rank I, II, III, and IV respectively. To estimate overall performance, these allotted grades were added to select the best model among all.

All four models used in this study are one-parameter, so comparison among them is useful and gives the correct information about the best model. Utilizing the rainfall-runoff data for all watersheds, this section compares results of the existing SCS-CN model (M_i), Woodward et al. model (M_ii), and Ajmal et al. model (M_iii) with the newly proposed simplified form of hybrid model (M_iv). To depict and evaluate the model performance, the comparison of models was first carried out by RMSE, NSE, PBIAS, n(t), and r² value for individual watersheds, then by examining cumulative performance of models for all watersheds by cumulative frequency curve, Box and Whisker plots and lastly by their total score obtained from analysis of the ranking and grading system (RGS). The best possible results of RMSE, NSE, and PBIAS values after carrying optimization are mentioned in Table 3 for all 114 US watersheds. The WS ID and data with bold values have been used for validation.

A scatter plot was drawn to compare RMSE values of all watersheds (Figure 3(a). Based on the cumulative frequency curve of RMSE value, Figure 4(a), the hybrid model performance was the best and its values on each segment were lowest. The Box and Whisker plot show (Figure 5(a)) that for both calibration and validation dataset, the RMSE value is minimum of M_iv model compared to other models. The cumulative mean RMSE value for M_i, M_ii, M_iii, and M_iv models was found as 6.76, 6.25, 6.01, and 5.71 mm, respectively for calibration datasets while it was found as 6.98, 6.27, 5.98, and 5.60 mm, respectively for validation datasets. Similarly, the median value also followed the same trend. The interquartile range is minimum and best for M_iv model (4.35–6.61), followed by M_iii model (4.87–7.02), M_ii model (5.32–7.35), and M_i model (5.83–8.30) in case of tested watersheds. For all watersheds, RMSE value was found to be lower than the conventional SCS-CN model. The standard deviation (SD) and standard error (SE) values are also less. The minimum value of RMSE shows that the new hybrid model performance was found best among all.

The variation in NSE values for different models for all watersheds are presented in (Figure 3(b). The NSE value was found negative for some watersheds. In model M_i, M_ii, and M_iii such negative NSE existed in 8, 4, and 2 watersheds. There was no such watershed found for M_iv model. When computed and observed runoff variance is more than the observed data variance, the NSE value can be negative. The overall mean NSE value for M_iv model is 0.74 which was found superior to other models, i.e. 0.56, 0.65, and 0.69 for M_i, M_ii, and M_iii model respectively. The cumulative frequency curve is always found on the upper side for the M_iv model which is an indicator of better performance (Figure 4(b)). According to the criteria defined by Ritter & Muñoz-Carpena (2013), 62, 80, 87, and 96 watersheds satisfactorily performed (NSE >0.65) for M_i, M_ii, M_iii, and M_iv models respectively. While NSE values greater than 0.75 were found for 32, 46, 52, and 61 watersheds for M_i, M_ii, M_iii, and M_iv models respectively. For tested watersheds, 9, 14, 15, and 21 watersheds found with NSE ≥0.75 and performed very well successively for M_i, M_ii, M_iii, and M_iv models. Figure 5(b) illustrates model performance by Box and Whisker plot which shows that the interquartile range of NSE was highest for M_iv model as 0.71–0.81 followed by M_iii (0.66–0.81), M_ii (0.63–0.79), and M_i (0.56–0.76) for calibrated watersheds and 0.67–0.80 of M_iv model followed by M_iii (0.63–0.78), M_ii (0.61–0.77), and M_i (0.50–0.75) for validated watersheds. To assess the impact of watershed size on NSE of the model, NSE values were plotted in a semi-log scale with the watershed area (Figure 6) but there no clear evidence was found to relate model performance with watershed size. Overall, the M_iv model was the best among all, and model M_iii results were better than model M_ii, whereas model M_i depicted the poorest runoff estimation with lower efficiency for most of the watersheds.

Performance based on PBIAS values is exhibited in Figures 3(c) and 4(c). By adopting the evaluation criteria mentioned in the ‘Model's Goodness-of-fit evaluation’ section, the M_i, M_ii, M_iii, and M_iv models showed very good performance (<± 10%) in 11, 27, 39, and 94 watersheds; good (±10% ≤ PBIAS < ±15%) in 21, 27, 37, and 8; satisfactory (±15% ≤ PBIAS < ±25%) in 53, 44, 22, and 6; unsatisfactory (PBIAS ≥ ±25%) in 29, 16, 16, and 6 watersheds, respectively. The M_iv model depicts consistent and improved PBIAS values in either a good or very good range over other models. Figure 5(c) shows PBIAS based Box and Whisker plot and it is evident that these values are going towards or closer to zero for the M_iv model for both calibrated and tested watersheds. The cumulative distribution curve depicted for the M_i, M_ii, and M_iii model underestimates the runoff by showing a positive PBIAS value but for the M_iv model 12 watersheds overestimate and the remaining 102 watersheds underestimate the runoff. Overall for the M_iv model, PBIAS is lowest (either positive or negative) of all models, which is a sign of better runoff prediction.

The n(t)-based assessment is done to check the perfection of a model for efficient runoff computation. Figures 3(d)–5(d) show the performance of all models according to n(t) values. The mean and median both value was found lowest for SCS-CN model as 0.78 and 0.75 respectively. For M_iv model n(t) value was best with the mean and median values as 1.16 and 1.06 respectively. Unsatisfactory performance was found in just 16 watersheds with the M_iv model followed by 27, 33, and 52 watersheds for the M_iii, M_ii, and M_i models. A good or very good performance was found in 38 watersheds using the M_iv model. For the M_iii, M_ii, and M_i models such performances were found in 32, 27, and 13 watersheds respectively.

The resulting cumulative value of minimum, maximum, mean, median, SD, SE, and interquartile range of RMSE, NSE, PBIAS, and n(t) of calibrated and tested watersheds for all four models are mentioned in Table 4.

By taking Senbeta et al. (1999) criteria as the yardstick of model improvement, r² was calculated to show the performance improvement of M_ii, M_iii, and M_iv models on the basis of the SCS-CN model (M_i). Based on visual analysis illustrated in Figure 7 the proposed hybrid model M_iv was encouraging for noteworthy improvement in runoff prediction on 110 watersheds. WS ID 19008, 19011, 44007, and 44012 are the only watersheds that have shown the opposite response by 14.79, 2.01, 10.83, and 6.33% respectively. For more than 99 watersheds (87%) of the total, M_iv performance was significantly better than M_ii and M_iii models. All models (M_ii, M_iii, and M_iv) revealed the improvement in performance over conventional SCS-CN model (M_i) but the rate of improvement is quite higher for model M_iv in maximum watersheds followed by models M_iii and M_ii. In addition, adopting the performance rating criteria (r² > 10%), M_ii, M_iii, and M_iv illustrate the improvement in their results of 87, 97, and 97 watersheds, respectively. Based on the improvement in model performance, the proposed hybrid model (M_iii and M_iv) performed better and indicated the importance of incorporating the SCS-CN model with the Ajmal (M_iii) model.

The performance of the calibrated hybrid model was tested on 36 watersheds. If we compare performance with the other three models, the hybrid model shows improvement in results of 29 watersheds while 7 watersheds predicted better runoff with Ajmal model (M_iii). It means both model performances were better than the SCS-CN model. The median value of RMSE for M_iv model was found as 5.66 mm while the M_iii, M_ii, and M_i models' values were 5.97, 6.36, and 6.98 mm respectively (Figure 5(a)). The NSE values for M_i, M_ii, M_iii, and M_iv model were found as 0.63, 0.69, 0.72, and 0.77 respectively (Figure 5(b)). Similarly, the PBIAS value for the M_iv model was best as 6.12% while its value for the M_iii, M_ii, and M_i model was found as successively 15.26, 17.76, and 22.32% (Figure 5(c)). For the tested catchments, model M_iv shows 33.51% average improvement while the M_iii model shows 25.56% average improvement in NSE values.

Like the statistical indices, model performance was also evaluated on the perfect-fit (1:1) line criteria by drawing a scatter plot between computed and observed runoff. Figure 8 illustrates representative fits for all model applications to watershed ID-13008, 26030, and 61001. The figure exhibits that the computed runoff values by M_iv model are closer to observed runoff values for the maximum number of events than other models. Likewise, it is seen that most of the data points in M_iv and M_iii models are closer to the line of best fit than M_ii and M_i models. Based on perfect fit, the M_iv model improves consistency in prediction, and performance was found superior followed by M_iii, M_ii, and M_i models. The SCS-CN-based M_i and M_ii models show inconsistent performance.

The model performance based on RGS and an overall score based on RGS is shown in Figure 9. Model M_iv attains rank 1 (score 4) in 102 watersheds, rank 2 (score 3) in 3 watersheds, rank 3 (score 2) in 5 watersheds, and rank 4 (score 1) in 4 watersheds. This shows M_iv got highest marks 431 and Rank 1 (102 × 4 + 3 × 3 + 5 × 2 + 4 × 1 = 431) followed by M_iii with 354 marks (Rank 2), M_ii with 237 marks (Rank 3), and M_i with only 118 marks (Rank 4) out of maximum 456 marks. Thus, based on the overall score based on RGS, model M_iv rated best and the order of model from best to poorest is found to be M_iv > M_iii > M_ii > M_i.

Catchment size based statistical evaluation has been made to check the consistency of model performance according to the size of watershed. The details of the result are mention in Table 5 and illustrated in Figure 10. The watersheds are categorized into five groups as C₁ = less than 1 ha (N = 15), C₂ = 1–10 ha (N = 56), C₃ = 10–100 ha (N = 19), C₄ = 100–1,000 ha (N = 17), and C₅ = greater than 1,000 ha (N = 7) watersheds. According to this, for each size range, the hybrid model M_iv consistently performs well and is followed by the M_iii, M_ii, and M_i model. The RMSE values were found to be 5.22, 5.87, 6.09, 4.66, and 6.41; NSE values 0.72, 0.76, 0.65, 0.79, and 0.74; PBIAS values 10.02, 5.22, 6.17, 4.76, and 1.74; and n(t) values 1.02, 1.2, 0.87, 1.49, and 1.05 of M_iv model for classified dataset C₁, C₂, C₃, C₄, and C₅ respectively. All performance indices were found superior for the C₄ and C₂ dataset. Based on the overall judgment of all statistical indices, M_iv is again considered the best model followed by the M_iii, M_ii, and M_i models.